# Basic notions reading group — Fall 2013

## Organization

- Organization
- All participants, but questions may be directed to Johan Commelin
- Location
- Hilbertruimte
- Time
- Mondays, 14.00–15.00

## Description

We will treat some topics (mostly categorical) that are useful for algebraic topologists and algebraic geometers. The focus will lie on Grothendieck topoi, and related notions. Besides that we will deal with some more algebraic concepts, aimed at cohomology.

A provisional list of topics:

- Sites and Grothendieck topoi, sheafification;
- Tannaka duality,
`G`-sets;
- Sheaves of abelian groups form an abelian category;
- Enough injectives in a suitable abelian category;
- Geometric morphisms, classifying topoi;
- Internal categories, internal sites/sheaves/topoi.

## Schedule

- 7 Oct 2013
- Introduction. Quick review of: sheaves on Open
`(X)`; equivalence with étalé bundles. Definition of coverages and Grothendieck topology. Sheaves on a site (definition relative to covarages, relative to sieves, and their equivalence).
- 14 Oct 2013
- Johan Commelin — Sheafification on a site (definition, properties: adjunction, idempotence, exactness). [Notes]
- 21 Oct 2013
- Frank Roumen — Grothendieck topoi: examples and basic properties. [Notes]
- 28 Oct 2013
- Joost Nuiten — Commutative group objects in an elementary topos form an abelian category.
- 4 Nov 2013
- No meeting.
- 11 Nov 2013
- Joost Nuiten — Commutative group objects in a Grothendieck topos have enough injectives.
- 18 Nov 2013
- Giovanni Caviglia — Geometric morphisms. [Notes]
- 25 Nov 2013
- Frank Roumen — Points of a topos.

## Literature

We will be using several sources as reference material.

- Artin, Michael; Alexander Grothendieck, Jean-Louis Verdier, eds. (1972).
*Séminaire de Géométrie Algébrique du Bois Marie - 1963–64 - Théorie des topos et cohomologie étale des schémas - (SGA 4)*
- Grothendieck, Alexander (1957), “Sur quelques points d'algèbre homologique”,
*The Tohoku Mathematical Journal. Second Series 9*: 119–221
- Mac Lane, Saunders; Moerdijk, Ieke (1994)
*Sheaves in geometry and logic. A first introduction to topos theory.*

## Participants

- Giovanni Caviglia
- Johan Commelin
- Joost Nuiten
- Frank Roumen
- Stephan Spahn
- Qizheng Yin

If you would like to participate in this seminar, please send me an email. (My email address is on my homepage.)